And I am talking of large companies like Ingram Micro, SIS, and the like ..
Well I've been in the PC component business and now in the import and distribution business, also dealt with Ingram Micro too...though that is all irrelevant, if you have studied business law, you will know better!
If the delivery charges are not on the bill, you will have a reasonable amount of time to correct it, as mistakes do happen.
In the OP's case he's got no worries at all if its the suppliers first communication in 6 months! We do not know anything of this case but the laws have it covered and it's mostly all commonsense and reasonable time frames.
Nobody knows what is going on. Maybe the supplier has been asking him to payup every week, though they would have to do it by mail, if they want to win the case and provide a separate invoice for transportation costs which is also a possibility that maybe denied..
It doesn't matter that the OP's customers case is closed, that is irrelevant. The person booking the delivery is the one that must pay, and that i suspect is the machine owner, who booked the transportation of the machine and delivered the machine to the OP's customer.
It could be a costly amount if its print machinery, in which case the transportation costs wouldn't be known until the job was done. So the OP only got the machine cost on the invoice for his customer, the OP gets the commission.
The OP should be aware he's got transportation costs to pay, though he's already got the invoice for the machine. We don't know the agreement between the two.
My guess is the OP is pulling a fast one and is ignoring the transport bill, for how long we don't really know.
If that is the case that the OP is trying it on, he will lose out...if there has been previous communication about transport costs. His invoice for the machine may not bear all the costs, as i suspect it's a heavy job, a lift job that takes time for packaging and delivery. Like print machines so there will be a separate invoice.
My guess is also that the OP has been working has a 3rd party salesman, and the supplier has agreed some terms with him on transportation costs.
I'm a perfectionist with ethics so it wouldn't happen to me

Reputation is everything, though some Singaporeans are still thinking short term gains are better.
I have only one thing to say about business people in Singapore and that is to study business law and international laws, otherwise you may bump into me and end up on the losing end of the line has two heavy weights in Singapore have already found that out, and dragged it has far as it would go with their insurance companies and litigation, before settling with me out of court.
International business law is complex and Singapore business locals are no different to any other small business people trying to make their living. It's only when they meet someone that does business by the laws that are laid down, that they become worried, and will do their best to bluff their way out of accountability.
I'm a pro at what i do, as an export consultant I would be held liable for taking your products into another country, that's what I'm hired for, to ensure companies exporting don't cock up with technical hindrances like hidden tax, custom law's ,AVA, NEA and other formalities it's my business to have a certain understanding of international laws and local business laws.
I'm the one that would be checking your financials, without your knowledge, to see if you are worthy of becoming a reliable distributor, I know what is right and what is wrong in every market legally and if i don't i have to find out, and weigh up the opportunity cost benefits. So i have it pretty well covered in all, including those dummies in Sim Lim.
And I'm pretty sure SE with his work too includes digging up what he needs to know, to be on the safe side in project work.
So I guess my expertise is being a spy and knowing every move legally and illegally in the market place, that's what I'm hired to find out. As business is about market share and taking market share, identifying weaknesses and strengths is all part of the package.
Singapore is a great place for doing business even more, if you pull on the government organisations to work with. The OP or the supplier are just other obstacles to deal with in business, no big deal if you know some basics of the laws but ethics will keep you in the good books much longer than bad eithics and short term gains!